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A binary parenthesization specifies the order of binary operations on a fixed ordered list of operands, combining exactly two existing expressions at each step. It has a full binary tree as its expression structure: operands are leaves and operations are internal vertices. For four operands $a,b,c,d$, the five possibilities are $((ab)c)d$, $(a(bc))d$, $(ab)(cd)$, $a((bc)d)$, and $a(b(cd))$.

A binary parenthesization of $n+1$ operands contains $n$ binary operations. At the last operation, its left expression can contain $k$ operations and its right expression $n-1-k$. The Catalan recurrence therefore gives $C_n$ possible evaluation structures. The order of operands stays fixed; this is not a count of their permutations. Even for an associative operation whose result is unchanged, different structures can have different evaluation costs, as in the matrix-chain multiplication problem.^[ref]

Interval dynamic programming stores answers for contiguous index intervals $[i,j]$ and combines shorter intervals. It applies when an optimal object can be split at an internal index while keeping subproblems contiguous.

For matrix-chain multiplication, with matrix dimensions $d_{i-1}\times d_i$, define $C(i,i)=0$ andThe final term is the scalar multiplication cost of combining the two products. Filling intervals in increasing length makes all dependencies available, giving $O(n^3)$ time and $O(n^2)$ storage. The recurrence optimizes parenthesization while retaining the original matrix order.

Both divide-and-conquer algorithms and dynamic programming derive answers from smaller problems. A typical divide-and-conquer algorithm, such as merge sort, splits an instance into separate parts and combines their answers. Dynamic programming additionally identifies repeated states across different decompositions and reuses their answers; matrix-chain multiplication is a standard example.

Thus a recursion tree may repeatedly solve the same state, whereas a dynamic programming table or memoization stores it once. The resulting recurrence and dependency order determine the running time. The principle of optimality supports combining optimal state answers; it does not by itself justify a greedy algorithm.