Loosely Tree-Based Alignment for Machine Translation
Daniel Gildea
University of Pennsylvania
dgildea@cis.upenn.edu
Abstract
We augment a model of translation based
on re-ordering nodes in syntactic trees in
order to allow alignments not conforming
to the original tree structure, while keep-
ing computational complexity polynomial
in the sentence length. This is done by
adding a new subtree cloning operation to
either tree-to-string or tree-to-tree align-
ment algorithms.
1 Introduction
Systems for automatic translation between lan-
guages have been divided into transfer-based ap-
proaches, which rely on interpreting the source
string into an abstract semantic representation
from which text is generated in the target lan-
guage, and statistical approaches, pioneered by
Brown et al. (1990), which estimate parameters for
a model of word-to-word correspondences and word
re-orderings directly from large corpora of par-
allel bilingual text. Only recently have hybrid
approaches begun to emerge, which apply prob-
abilistic models to a structured representation of
the source text. Wu (1997) showed that restrict-
ing word-level alignments between sentence pairs
to observe syntactic bracketing constraints signif-
icantly reduces the complexity of the alignment
problem and allows a polynomial-time solution.
Alshawi et al. (2000) also induce parallel tree struc-
tures from unbracketed parallel text, modeling the
generation of each node’s children with a finite-state
transducer. Yamada and Knight (2001) present an
algorithm for estimating probabilistic parameters for
a similar model which represents translation as a se-
quence of re-ordering operations over children of
nodes in a syntactic tree, using automatic parser out-
put for the initial tree structures. The use of explicit
syntactic information for the target language in this
model has led to excellent translation results (Ya-
mada and Knight, 2002), and raises the prospect of
training a statistical system using syntactic informa-
tion for both sides of the parallel corpus.
Tree-to-tree alignment techniques such as prob-
abilistic tree substitution grammars (Hajiˇc et al.,
2002) can be trained on parse trees from parallel
treebanks. However, real bitexts generally do not
exhibit parse-tree isomorphism, whether because of
systematic differences between how languages ex-
press a concept syntactically (Dorr, 1994), or simply
because of relatively free translations in the training
material.
In this paper, we introduce “loosely” tree-based
alignment techniques to address this problem. We
present analogous extensions for both tree-to-string
and tree-to-tree models that allow alignments not
obeying the constraints of the original syntactic tree
(or tree pair), although such alignments are dispre-
ferred because they incur a cost in probability. This
is achieved by introducing a clone operation, which
copies an entire subtree of the source language syn-
tactic structure, moving it anywhere in the target
language sentence. Careful parameterization of the
probability model allows it to be estimated at no ad-
ditional cost in computational complexity. We ex-
pect our relatively unconstrained clone operation to
allow for various types of structural divergence by
providing a sort of hybrid between tree-based and
unstructured, IBM-style models.
We first present the tree-to-string model, followed
by the tree-to-tree model, before moving on to align-
ment results for a parallel syntactically annotated
Korean-English corpus, measured in terms of align-
ment perplexities on held-out test data, and agree-
ment with human-annotated word-level alignments.
2 The Tree-to-String Model
We begin by summarizing the model of
Yamada and Knight (2001), which can be thought
of as representing translation as an Alexander
Calder mobile. If we follow the process of an
English sentence’s transformation into French,
the English sentence is first given a syntactic tree
representation by a statistical parser (Collins, 1999).
As the first step in the translation process, the
children of each node in the tree can be re-ordered.
For any node with m children, m! re-orderings are
possible, each of which is assigned a probability
P
order
conditioned on the syntactic categories of
the parent node and its children. As the second
step, French words can be inserted at each node
of the parse tree. Insertions are modeled in two
steps, the first predicting whether an insertion to
the left, an insertion to the right, or no insertion
takes place with probability P
ins
, conditioned on
the syntactic category of the node and that of its
parent. The second step is the choice of the inserted
word P
t
(fjNULL), which is predicted without
any conditioning information. The final step, a
French translation of each original English word,
at the leaves of the tree, is chosen according to a
distribution P
t
(fje). The French word is predicted
conditioned only on the English word, and each
English word can generate at most one French
word, or can generate a NULL symbol, representing
deletion. Given the original tree, the re-ordering,
insertion, and translation probabilities at each node
are independent of the choices at any other node.
These independence relations are analogous to those
of a stochastic context-free grammar, and allow for
efficient parameter estimation by an inside-outside
Expectation Maximization (EM) algorithm. The
computation of inside probabilities  , outlined
below, considers possible reordering of nodes in the
original tree in a bottom-up manner:
for all nodes "
i
in input tree T do
for all k;l such that 1 <k<l<Ndo
for all orderings  of the children "
1
:::"
m
of "
i
do
for all partitions of span k;l into k
1
;l
1
:::k
m
;l
m
do
 ("
i
;k;l)+= P
order
( j"
i
)
Q
m
j=1
 ("
j
;k
j
;l
j
)
end for
end for
end for
end for
This algorithm has computational complexity
O(jTjN
m+2
), where m is the maximum number of
children of any node in the input tree T, and N
the length of the input string. By storing partially
completed arcs in the chart and interleaving the in-
ner two loops, complexity of O(jTjn
3
m!2
m
) can be
achieved. Thus, while the algorithm is exponential
in m, the fan-out of the grammar, it is polynomial in
the size of the input string. Assuming jTj = O(n),
the algorithm is O(n
4
).
The model’s efficiency, however, comes at a cost.
Not only are many independence assumptions made,
but many alignments between source and target sen-
tences simply cannot be represented. As a minimal
example, take the tree:
A
B
X Y
Z
Of the six possible re-orderings of the three ter-
minals, the two which would involve crossing the
bracketing of the original tree (XZY and YZX)
are not allowed. While this constraint gives us a
way of using syntactic information in translation,
it may in many cases be too rigid. In part to deal
with this problem, Yamada and Knight (2001) flat-
ten the trees in a pre-processing step by collapsing
nodes with the same lexical head-word. This allows,
for example, an English subject-verb-object (SVO)
structure, which is analyzed as having a VP node
spanning the verb and object, to be re-ordered as
VSO in a language such as Arabic. Larger syntactic
divergences between the two trees may require fur-
ther relaxation of this constraint, and in practice we
expect such divergences to be frequent. For exam-
ple, a nominal modifier in one language may show
up as an adverbial in the other, or, due to choices
such as which information is represented by a main
verb, the syntactic correspondence between the two
S
VP
NP
NNC
NNC
Kyeo-ul
PAD
e
PAU
Neun
VP
NP
NNC
NNC
Su-Kap
PCA
eul
VP
NP
NNU
Myeoch
NNX
NNX
Khyeol-Re
XSF
Ssik
VP
NP
NNC
Ci-Keup
LV
VV
VV
Pat
EFN
Ci
S
VP
VP
VP
VP
NP
NNC
Ci-Keup
NULL
LV
VV
VV
Pat
NULL
EFN
Ci
NULL
NP
NNU
Myeoch
how
NNX
XSF
Ssik
many
NNX
Khyeol-Re
pairs
NP
NNC
NNC
Su-Kap
gloves
PCA
eul
NULL
NP
VP
LV
VV
VV
Pat
each
EFN
Ci
you
NP
NNC
Ci-Keup
issued
NNC
PAD
e
in
PAU
Neun
NULL
NNC
Kyeo-ul
winter
Figure 1: Original Korean parse tree, above, and transformed tree after reordering of children, subtree
cloning (indicated by the arrow), and word translation. After the insertion operation (not shown), the tree’s
English yield is: How many pairs of gloves is each of you issued in winter?
sentences may break down completely.
2.1 Tree-to-String Clone Operation
In order to provide some flexibility, we modify the
model in order to allow for a copy of a (translated)
subtree from the English sentences to occur, with
some cost, at any point in the resulting French sen-
tence. For example, in the case of the input tree
A
B
X Y
Z
a clone operation making a copy of node 3 as a new
child of B would produce the tree:
A
B
X Z Y
Z
This operation, combined with the deletion of the
original node Z, produces the alignment (XZY)
that was disallowed by the original tree reorder-
ing model. Figure 1 shows an example from our
Korean-English corpus where the clone operation al-
lows the model to handle a case of wh-movement in
the English sentence that could not be realized by
any reordering of subtrees of the Korean parse.
The probability of adding a clone of original node
"
i
as a child of node "
j
is calculated in two steps:
first, the choice of whether to insert a clone under
"
j
, with probability P
ins
(clonej"
j
), and the choice
of which original node to copy, with probability
P
clone
("
i
jclone =1)=
P
makeclone
("
i
)
P
k
P
makeclone
("
k
)
where P
makeclone
is the probability of an original
node producing a copy. In our implementation, for
simplicity, P
ins
(clone) is a single number, estimated
by the EM algorithm but not conditioned on the par-
ent node "
j
, and P
makeclone
is a constant, meaning
that the node to be copied is chosen from all the
nodes in the original tree with uniform probability.
It is important to note that P
makeclone
is not de-
pendent on whether a clone of the node in ques-
tion has already been made, and thus a node may
be “reused” any number of times. This indepen-
dence assumption is crucial to the computational
tractability of the algorithm, as the model can be
estimated using the dynamic programming method
above, keeping counts for the expected number of
times each node has been cloned, at no increase in
computational complexity. Without such an assump-
tion, the parameter estimation becomes a problem
of parsing with crossing dependencies, which is ex-
ponential in the length of the input string (Barton,
1985).
3 The Tree-to-Tree Model
The tree-to-tree alignment model has tree transfor-
mation operations similar to those of the tree-to-
string model described above. However, the trans-
formed tree must not only match the surface string
of the target language, but also the tree structure as-
signed to the string by the treebank annotators. In or-
der to provide enough flexibility to make this possi-
ble, additional tree transformation operations allow
a single node in the source tree to produce two nodes
in the target tree, or two nodes in the source tree to
be grouped together and produce a single node in
the target tree. The model can be thought of as a
synchronous tree substitution grammar, with proba-
bilities parameterized to generate the target tree con-
ditioned on the structure of the source tree.
The probability P(T
b
jT
a
) of transforming the
source tree T
a
into target tree T
b
is modeled in a
sequence of steps proceeding from the root of the
target tree down. At each level of the tree:
1. At most one of the current node’s children is
grouped with the current node in a single ele-
mentary tree, with probability P
elem
(t
a
j"
a
)
children("
a
)), conditioned on the current
node "
a
and its children (ie the CFG produc-
tion expanding "
a
).
2. An alignment of the children of the current
elementary tree is chosen, with probability
P
align
( j"
a
) children(t
a
)). This alignment
operation is similar to the re-order operation
in the tree-to-string model, with the extension
that 1) the alignment  can include insertions
and deletions of individual children, as nodes
in either the source or target may not corre-
spond to anything on the other side, and 2) in
the case where two nodes have been grouped
into t
a
, their children are re-ordered together in
one step.
In the final step of the process, as in the tree-to-
string model, lexical items at the leaves of the tree
are translated into the target language according to a
distribution P
t
(fje).
Allowing non-1-to-1 correspondences between
nodes in the two trees is necessary to handle the
fact that the depth of corresponding words in the
two trees often differs. A further consequence of
allowing elementary trees of size one or two is that
some reorderings not allowed when reordering the
children of each individual node separately are now
possible. For example, with our simple tree
A
B
X Y
Z
if nodes A and B are considered as one elementary
tree, with probability P
elem
(t
a
jA ) BZ), their col-
lective children will be reordered with probability
P
align
(f(1; 1)(2; 3)(3; 2)gjA ) XYZ)
A
X Z Y
giving the desired word ordering XZY. However,
computational complexity as well as data sparsity
prevent us from considering arbitrarily large ele-
mentary trees, and the number of nodes considered
at once still limits the possible alignments. For ex-
ample, with our maximum of two nodes, no trans-
formation of the tree
A
B
W X
C
Y Z
is capable of generating the alignment WYXZ.
In order to generate the complete target tree, one
more step is necessary to choose the structure on the
target side, specifically whether the elementary tree
has one or two nodes, what labels the nodes have,
and, if there are two nodes, whether each child at-
taches to the first or the second. Because we are
ultimately interested in predicting the correct target
string, regardless of its structure, we do not assign
probabilities to these steps. The nonterminals on the
target side are ignored entirely, and while the align-
ment algorithm considers possible pairs of nodes as
elementary trees on the target side during training,
the generative probability model should be thought
of as only generating single nodes on the target side.
Thus, the alignment algorithm is constrained by the
bracketing on the target side, but does not generate
the entire target tree structure.
While the probability model for tree transforma-
tion operates from the top of the tree down, prob-
ability estimation for aligning two trees takes place
by iterating through pairs of nodes from each tree in
bottom-up order, as sketched below:
for all nodes "
a
in source tree T
a
in bottom-up order do
for all elementary trees t
a
rooted in "
a
do
for all nodes "
b
in target tree T
b
in bottom-up order do
for all elementary trees t
b
rooted in "
b
do
for all alignments  of the children of t
a
and t
b
do
 ("
a
;"
b
) +=
P
elem
(t
a
j"
a
)P
align
( j"
i
)
Q
(i;j)2 
 ("
i
;"
j
)
end for
end for
end for
end for
end for
The outer two loops, iterating over nodes in each
tree, require O(jTj
2
). Because we restrict our el-
ementary trees to include at most one child of the
root node on either side, choosing elementary trees
for a node pair is O(m
2
), where m refers to the max-
imum number of children of a node. Computing the
alignment between the 2m children of the elemen-
tary tree on either side requires choosing which sub-
set of source nodes to delete, O(2
2m
), which subset
of target nodes to insert (or clone), O(2
2m
), and how
to reorder the remaining nodes from source to target
tree, O((2m)!). Thus overall complexity of the algo-
rithm is O(jTj
2
m
2
4
2m
(2m)!), quadratic in the size
of the input sentences, but exponential in the fan-out
of the grammar.
3.1 Tree-to-Tree Clone Operation
Allowing m-to-n matching of up to two nodes
on either side of the parallel treebank allows for
limited non-isomorphism between the trees, as in
Hajiˇc et al. (2002). However, even given this flexi-
bility, requiring alignments to match two input trees
rather than one often makes tree-to-tree alignment
more constrained than tree-to-string alignment. For
example, even alignments with no change in word
order may not be possible if the structures of the
two trees are radically mismatched. This leads us
to think it may be helpful to allow departures from
Tree-to-String Tree-to-Tree
elementary tree grouping P
elem
(t
a
j"
a
) children("
a
))
re-order P
order
( j" ) children(")) P
align
( j"
a
) children(t
a
))
insertion P
ins
(left; right; nonej")  can include “insertion” symbol
lexical translation P
t
(fje) P
t
(fje)
with cloning P
ins
(clonej")  can include “clone” symbol
P
makeclone
(") P
makeclone
(")
Table 1: Model parameterization
the constraints of the parallel bracketing, if it can
be done in without dramatically increasing compu-
tational complexity.
For this reason, we introduce a clone operation,
which allows a copy of a node from the source tree to
be made anywhere in the target tree. After the clone
operation takes place, the transformation of source
into target tree takes place using the tree decomposi-
tion and subtree alignment operations as before. The
basic algorithm of the previous section remains un-
changed, with the exception that the alignments  
between children of two elementary trees can now
include cloned, as well as inserted, nodes on the tar-
get side. Given that  specifies a new cloned node
as a child of "
j
, the choice of which node to clone is
made as in the tree-to-string model:
P
clone
("
i
jclone 2  )=
P
makeclone
("
i
)
P
k
P
makeclone
("
k
)
Because a node from the source tree is cloned with
equal probability regardless of whether it has al-
ready been “used” or not, the probability of a clone
operation can be computed under the same dynamic
programming assumptions as the basic tree-to-tree
model. As with the tree-to-string cloning operation,
this independence assumption is essential to keep
the complexity polynomial in the size of the input
sentences.
For reference, the parameterization of all four
models is summarized in Table 1.
4 Data
For our experiments, we used a parallel Korean-
English corpus from the military domain (Han et al.,
2001). Syntactic trees have been annotated by hand
for both the Korean and English sentences; in this
paper we will be using only the Korean trees, mod-
eling their transformation into the English text. The
corpus contains 5083 sentences, of which we used
4982 as training data, holding out 101 sentences for
evaluation. The average Korean sentence length was
13 words. Korean is an agglutinative language, and
words often contain sequences of meaning-bearing
suffixes. For the purposes of our model, we rep-
resented the syntax trees using a fairly aggressive
tokenization, breaking multimorphemic words into
separate leaves of the tree. This gave an average
of 21 tokens for the Korean sentences. The aver-
age English sentence length was 16. The maximum
number of children of a node in the Korean trees
was 23 (this corresponds to a comma-separated list
of items). 77% of the Korean trees had no more
than four children at any node, 92% had no more
than five children, and 96% no more than six chil-
dren. The vocabulary size (number of unique types)
was 4700 words in English, and 3279 in Korean —
before splitting multi-morphemic words, the Korean
vocabulary size was 10059. For reasons of compu-
tation speed, trees with more than 5 children were
excluded from the experiments described below.
5 Experiments
We evaluate our translation models both in terms
agreement with human-annotated word-level align-
ments between the sentence pairs. For scoring
the viterbi alignments of each system against gold-
standard annotated alignments, we use the alignment
error rate (AER) of Och and Ney (2000), which
measures agreement at the level of pairs of words:
1
AER =1−
2jA \ Gj
jAj + jGj
1
While Och and Ney (2000) differentiate between sure and
possible hand-annotated alignments, our gold standard align-
ments come in only one variety.
Alignment
Error Rate
IBM Model 1 .37
IBM Model 2 .35
IBM Model 3 .43
Tree-to-String .42
Tree-to-String, Clone .36
Tree-to-String, Clone P
ins
= :5 .32
Tree-to-Tree .49
Tree-to-Tree, Clone .36
Table 2: Alignment error rate on Korean-English corpus
where A is the set of word pairs aligned by the au-
tomatic system, and G the set aligned in the gold
standard. We provide a comparison of the tree-based
models with the sequence of successively more com-
plex models of Brown et al. (1993). Results are
shown in Table 2.
The error rates shown in Table 2 represent the
minimum over training iterations; training was
stopped for each model when error began to in-
crease. IBM Models 1, 2, and 3 refer to
Brown et al. (1993). “Tree-to-String” is the model
of Yamada and Knight (2001), and “Tree-to-String,
Clone” allows the node cloning operation of Section
2.1. “Tree-to-Tree” indicates the model of Section 3,
while “Tree-to-Tree, Clone” adds the node cloning
operation of Section 3.1. Model 2 is initialized from
the parameters of Model 1, and Model 3 is initialized
from Model 2. The lexical translation probabilities
P
t
(fje) for each of our tree-based models are initial-
ized from Model 1, and the node re-ordering proba-
bilities are initialized uniformly. Figure 1 shows the
viterbi alignment produced by the “Tree-to-String,
Clone” system on one sentence from our test set.
We found better agreement with the human align-
ments when fixing P
ins
(left) in the Tree-to-String
model to a constant rather than letting it be deter-
mined through the EM training. While the model
learned by EM tends to overestimate the total num-
ber of aligned word pairs, fixing a higher probability
for insertions results in fewer total aligned pairs and
therefore a better trade-off between precision and
recall. As seen for other tasks (Carroll and Char-
niak, 1992; Merialdo, 1994), the likelihood crite-
rion used in EM training may not be optimal when
evaluating a system against human labeling. The
approach of optimizing a small number of metapa-
rameters has been applied to machine translation by
Och and Ney (2002). It is likely that the IBM mod-
els could similarly be optimized to minimize align-
ment error – an open question is whether the opti-
mization with respect to alignment error will corre-
spond to optimization for translation accuracy.
Within the strict EM framework, we found
roughly equivalent performance between the IBM
models and the two tree-based models when making
use of the cloning operation. For both the tree-to-
string and tree-to-tree models, the cloning operation
improved results, indicating that adding the flexibil-
ity to handle structural divergence is important when
using syntax-based models. The improvement was
particularly significant for the tree-to-tree model, be-
cause using syntactic trees on both sides of the trans-
lation pair, while desirable as an additional source of
information, severely constrains possible alignments
unless the cloning operation is allowed.
The tree-to-tree model has better theoretical com-
plexity than the tree-to-string model, being quadratic
rather than quartic in sentence length, and we found
this to be a significant advantage in practice. This
improvement in speed allows longer sentences and
more data to be used in training syntax-based mod-
els. We found that when training on sentences of up
60 words, the tree-to-tree alignment was 20 times
faster than tree-to-string alignment. For reasons of
speed, Yamada and Knight (2002) limited training
to sentences of length 30, and were able to use only
one fifth of the available Chinese-English parallel
corpus.
6 Conclusion
Our loosely tree-based alignment techniques allow
statistical models of machine translation to make use
of syntactic information while retaining the flexibil-
ity to handle cases of non-isomorphic source and tar-
get trees. This is achieved with a clone operation pa-
rameterized in such a way that alignment probabili-
ties can be computed with no increase in asymptotic
computational complexity.
We present versions of this technique both for
tree-to-string models, making use of parse trees for
one of the two languages, and tree-to-tree models,
which make use of parallel parse trees. Results in
terms of alignment error rate indicate that the clone
operation results in better alignments in both cases.
On our Korean-English corpus, we found roughly
equivalent performance for the unstructured IBM
models, and the both the tree-to-string and tree-to-
tree models when using cloning. To our knowl-
edge these are the first results in the literature for
tree-to-tree statistical alignment. While we did not
see a benefit in alignment error from using syntactic
trees in both languages, there is a significant practi-
cal benefit in computational efficiency. We remain
hopeful that two trees can provide more information
than one, and feel that extensions to the “loosely”
tree-based approach are likely to demonstrate this
using larger corpora.
Another important question we plan to pursue is
the degree to which these results will be borne out
with larger corpora, and how the models may be re-
fined as more training data is available. As one ex-
ample, our tree representation is unlexicalized, but
we expect conditioning the model on more lexical
information to improve results, whether this is done
by percolating lexical heads through the existing
trees or by switching to a strict dependency repre-
sentation.
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