Consider a binary tree labelled by some ordered set of letters. Traversing the tree in preorder determines a sequence of letters - a word.
A binary tree is called full if all its non-leaf vertices have exactly two children.
Given two such binary trees that are also full, and determine the same word, does it follow that the two trees are equal?
There will definitely be more than one full binary tree that will yield a particular pre-order traversal. So short answer to your question will be no.
Here is a sketch for a potential counter-example. Consider the following 2 trees.
There will exist an assignment of letters such that the pre-order traversal on both the above trees yields the same word. However, the trees are clearly dissimilar, and hence your claim is false.