  Lam, Lee and Shimozono introduced the structure of bumpless pipedreams in
their study of back stable Schubert calculus. They found that a specific family
of bumpless pipedreams, called EG-pipedreams, can be used to interpret the
Edelman-Greene coefficients appearing in the expansion of a Stanley symmetric
function in the basis of Schur functions. It is well known that the
Edelman-Greene coefficients can also be interpreted in terms of reduced word
tableaux for permutations. Lam, Lee and Shimozono proposed the problem of
finding a shape preserving bijection between reduced word tableaux for a
permutation $w$ and EG-pipedreams of $w$. In this paper, we construct such a
bijection. The key ingredients are two new developed isomorphic tree structures
associated to $w$: the modified Lascoux-Sch\"{u}tzenberger tree of $w$ and the
Edelman-Greene tree of $w$. Using the Little map, we show that the leaves in
the modified Lascoux-Sch\"{u}tzenberger of $w$ are in bijection with the
reduced word tableaux for $w$. On the other hand, applying the droop operation
on bumpless pipedreams also introduced by Lam, Lee and Shimozono, we show that
the leaves in the Edelman-Greene tree of $w$ are in bijection with the
EG-pipedreams of $w$. This allows us to establish a shape preserving one-to-one
correspondence between reduced word tableaux for $w$ and EG-pipedreams of $w$.
