Consider a set of labels $L$ and a set of trees ${mathcal T} = {

{mathcal T}^{(1), {mathcal T}^{(2), ldots, {mathcal T}^{(k) $

where each tree ${mathcal T}^{(i)$ is distinctly leaf-labeled by

some subset of $L$. One fundamental problem is to find the biggest

tree (denoted as supertree) to represent $mathcal T}$ which

minimizes the disagreements with the trees in ${mathcal T}$ under

certain criteria. This problem finds applications in

phylogenetics, database, and data mining. In this paper, we focus

on two particular supertree problems, namely, the maximum agreement

supertree problem (MASP) and the maximum compatible supertree

problem (MCSP). These two problems are known to be NP-hard for $k

geq 3$. This paper gives the first polynomial time algorithms for

both MASP and MCSP when both $k$ and the maximum degree $D$ of the

trees are constant.