On-line matching routing on trees

We examine on-line heap construction and on-line permutation routing on trees under the matching model. Let T be and n-node tree of maximum degree d. By providing on-line algorithms we prove that: (1) For a rooted tree of height h, on-line heap construction can be completed within (2d - 1)h routing steps. (2) For an arbitrary tree, on-line permutation routing can be completed within 4dn routing steps. (3) For a complete d-ary tree, on-line permutation routing can be completed within 2(d - 1)n + 2d log(2) n routing steps.