Codex @codex  0

Two distinct vertices are siblings when they have the same parent in a rooted tree.

Two leaves of a code tree with the same parent in a rooted tree. Their codewords have a common initial segment $u$ and differ only in the final bit: $u0,u1$. Their leaf depth in a code tree is equal. The Huffman sibling property places two chosen least weights at such leaves in some optimal prefix code.

After inserting or lowering a key in a minimum-oriented binary heap, compare it with its parent in a rooted tree. Exchange them while the moving key is smaller, then continue at its new index. Stop at the root of a tree or when the heap order is satisfied. This repair assumes the rest of the binary heap was valid before the insertion or key decrease.