The instructor introduces the topic of the video which is to compute the diameter of a binary tree.

• The instructor welcomes viewers and sets the agenda to write neat code for solving the diameter of a binary tree.
• The diameter of a binary tree is explained as the length of the longest path between any two nodes, which may or may not pass through the root.

An example tree is provided to illustrate the concept of tree diameter.

• The instructor uses an example tree to illustrate the concept of diameter, showing a path from node 4 to node 3 with a diameter of 3.
• Other potential diameters are discussed, indicating that the longest path does not always involve the root node.

The brute force approach to solving the problem is explained using examples.

• The instructor outlines a brute force approach by considering every node as the topmost node in the diameter and calculating the maximum path from there.
• It is noted that the brute force approach is O(n^2), where n is the number of nodes, due to repeated work.

The instructor transitions from the brute force approach to an introduction of the optimized solution.

• The instructor explains that by using recursion, the repeated work in the brute force approach can be eliminated.
• Instead of a top-down approach, a bottom-up approach is suggested where the diameter and height of the tree are computed starting from the leaf nodes.

Detailed explanation of the optimized recursive solution, including the calculation of diameter and height of subtrees.

• The instructor describes that each node in the tree will have a diameter and height associated with it.
• The concept of height is introduced and explained using the example tree, noting the convention that the height of a null tree is -1.
• The process of computing the diameter and height for each node is explained step by step, from the leaf nodes up to the root.

The instructor walks through the code implementation of the optimized solution.

• The instructor introduces a global result variable to keep track of the maximum diameter found during the depth-first search.
• A nested function, depth first search, is defined to calculate the height and update the result if a larger diameter is found.
• The base case for the recursive function is explained, and the return value of the height of null trees as -1 is justified.
• The recursive calls for left and right subtrees are shown in code, along with the calculation of the diameter and the updating of the result.