Classic DP Families: LCS, Matrix-Chain Multiplication, and Optimal BSTs

The previous lesson worked SRBOT on a one-dimensional subproblem family (rod cutting, indexed by a single length). This note applies the same recipe to three canonical two-dimensional DPs — longest common subsequence (LCS), matrix-chain multiplication, and optimal binary search trees — each indexed by a pair of indices (a substring or subarray range), which is the signature shape of DP problems over sequences or orderings. The optimal-BST problem closes a direct loop back to Topic 3's balanced-tree machinery, this time optimising expected search cost rather than worst-case height.