The postorder traversal of a binary tree is 8,9,6,7,4,5,2,3,1. The inorder traversal of the same tree is 8,6,9,4,7,2,5,1,3. The height of a tree is the length of the longest path from the root to any leaf. The height of the binary tree above is ________.

To find the height of the tree, we reconstruct the binary tree from the given traversals:

• Postorder: $8, 9, 6, 7, 4, 5, 2, 3, 1$

• Inorder: $8, 6, 9, 4, 7, 2, 5, 1, 3$

1. The last element in postorder is the root: 1.

2. In the inorder traversal, 1 splits the tree into:

   • Left subtree inorder: $[8, 6, 9, 4, 7, 2, 5]$

   • Right subtree inorder: $[3]$

3. From the postorder traversal:

   • Left subtree postorder: $[8, 9, 6, 7, 4, 5, 2]$ (root is 2)

   • Right subtree postorder: $[3]$ (leaf node at depth 1)

4. For root 2:

   • Inorder left: $[8, 6, 9, 4, 7]$, right: $[5]$

   • Postorder left: $[8, 9, 6, 7, 4]$ (root is 4), right: $[5]$

5. For root 4:

   • Inorder left: $[8, 6, 9]$, right: $[7]$

   • Postorder left: $[8, 9, 6]$ (root is 6), right: $[7]$

6. For root 6:

   • Inorder left: $[8]$, right: $[9]$

   • Postorder left: $[8]$, right: $[9]$

Now, let's trace the path from the root (node 1) to the deepest leaf (nodes 8 or 9):

The length of this path (number of edges from root to leaf) is $4$.

Hence, the height of the binary tree is 4.