#### Approach #1: Depth-First Search [Accepted]

Intuition

We can serialize each subtree. For example, the tree

   1
/ \
2   3
/ \
4   5


can be represented as the serialization 1,2,#,#,3,4,#,#,5,#,#, which is a unique representation of the tree.

Algorithm

Perform a depth-first search, where the recursive function returns the serialization of the tree. At each node, record the result in a map, and analyze the map after to determine duplicate subtrees.

Complexity Analysis

• Time Complexity: , where is the number of nodes in the tree. We visit each node once, but each creation of serial may take work.

• Space Complexity: , the size of count.

#### Approach #2: Unique Identifier [Accepted]

Intuition

Suppose we have a unique identifier for subtrees: two subtrees are the same if and only if they have the same id.

Then, for a node with left child id of x and right child id of y, (node.val, x, y) uniquely determines the tree.

Algorithm

If we have seen the triple (node.val, x, y) before, we can use the identifier we've remembered. Otherwise, we'll create a new one.

Complexity Analysis

• Time Complexity: , where is the number of nodes in the tree. We visit each node once.

• Space Complexity: . Every structure we use is using storage per node.

Analysis written by: @awice. Approach #2 inspired by @StefanPochmann.