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By rewriting the famous hook-content formula it easily follows that there are$\prod\limits_{1 \le i < j \le n} \frac{k_j - k_i + j -i}{j-i}$ semistandardtableaux of shape $(k_n,k_{n-1},...,k_1)$ with entries in $\{1,2,...,n\}$ or,equivalently, Gelfand-Tsetlin patterns with bottom row $(k_1,...,k_n)$. In thisarticle we introduce certain sequences of labeled trees, the signed enumerationof which is also given by this formula. In these trees, vertices as well asedges are labeled, the crucial condition being that each edge label liesbetween the vertex labels of the two endpoints of the edge. This notion enablesus to give combinatorial explanations of the shifted antisymmetry of theformula and its polynomiality. Furthermore, we propose to develop an analogapproach of combinatorial reasoning for monotone triangles and explain how thismay lead to a combinatorial understanding of the alternating sign matrixtheorem.

Sequences of labeled trees related to Gelfand-Tsetlin patterns