Find whether a tree is a subtree of other

Question

There are two binary trees T1 and T2 which store character data, duplicates allowed.
How can I find whether T2 is a subtree of T1 ? .
T1 has millions of nodes and

Answer 1

If you are memory/storage bound (i.e. can't pre-manipulate and store the trees in alternate forms), you probably won't be able to do anything better than the brute force search suggested by some other answers (traverse P1 looking for matching root of P2, traverse both to determine whether the node is in-fact the root of a matching sub-tree, continue with original traversal if it isn't a match). This search operates in O(n * m) where n is the size of P1 and m is the size of P2. With depth checks and other potential optimizations depending on the tree data you have available, this man be optimized a bit, but it's still generally O(n * m). Depending on your specific circumstances, this may be the only reasonable approach.

If you're not memory/storage bound and don't mind a bit of complexity, I believe this could be improved to O(n + m) by reducing to a longest common substring problem with the help of a generalized suffix tree. Some discussion on this for a similar problem can be found here. Maybe when I have more time, I'll come back and edit with more specifics on an implementation.