Sequences of labeled trees related to Gelfand-Tsetlin patterns

By rewriting the famous hook-content formula it easily follows that there are $\prod _{1\le i<j\le n}\frac{k_j-k_i+j-i}{j-i}$ semistandard tableaux of shape $(k_n,k_{n-1},\mathrm{.}\mathrm{.}\mathrm{.},k_1)$ with entries in $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}$ or, equivalently, Gelfand-Tsetlin patterns with bottom row $(k_1,\mathrm{.}\mathrm{.}\mathrm{.},k_n)$. In this article we introduce certain sequences of labeled trees, the signed enumeration of which is also given by this formula. In these trees, vertices as well as edges are labeled, the crucial condition being that each edge label lies between the vertex labels of the two endpoints of the edge. This notion enables us to give combinatorial explanations of the shifted antisymmetry of the formula and its polynomiality. Furthermore, we propose to develop an analog approach of combinatorial reasoning for monotone triangles and explain how this may lead to a combinatorial understanding of the alternating sign matrix theorem.