Let T be any binary tree with leaves S1, ..., Sl. Let W1, ..., We E R be an assignment of weights such that si receives weight Wi. For non-leaf vertices, we iteratively define the weight of a vertex to be the sum of the weights of their children. (See below for an example.) Prove that the average weighted depth of T is equal to the sum of the weights of non-leaf vertices of T. wi + W2+W3 + 24 + 25 + w6+ 27 W1 + W2 + W3 W4+ W5 + W6 + W7 W1 + W2 W3 W5 + w6+w7 W4 W6 + W7 W1 W2 W3 W5 W6 W7 (Note: Your proof must apply for all binary trees, not just the example given.)