Enumeration of weighted plane trees by a permutation model

This work addresses an enumeration problem on weighted bi-colored plane trees with prescribed vertex data, with all vertices labeled distinctly. We give a bijection proof of the enumeration formula originally due to Kochetkov, hence affirmatively answer a question of Adrianov-Pakovich-Zvonkin. The argument is purely combinatorial and totally constructive, remaining valid for real-valued edge weights. A central process is a geometric construction that directly encodes each tree as a permutation. We also exhibit algebraic relationships between the enumeration problem, the partial order on partitions of vertices and the Stirling numbers of the second kind. Some computation examples are presented as appendices.