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Let $\mathbb{K}$ be a field and $n$ be a positive integer. Let $\Gamma =([n],E)$ be a simple graph, where $[n]=\{1,\ldots, n\}$. If $S=\mathbb{K}[x_1,\ldots, x_n, y_1, \ldots, y_n]$ is a polynomial ring, then the graded ideal \[L_\Gamma^\mathbb{K}(2) = \left( x_{i}x_{j} + y_{i}y_{j} \colon \quad \{i, j\}\in E(\Gamma)\right) \subset S,\] is called the Lov\'{a}sz-Saks-Schrijverideal, LSS-ideal for short, of $\Gamma$ with respect to $\mathbb{K}$. In thepresent paper, we compute a Gr\"obner basis of this ideal with respect tolexicographic ordering induced by $x_1>\cdots>x_n>y_1>\cdots>y_n$ when$\Gamma=T$ is a tree. As a result, we show that it is independent of the choiceof the ground field $\mathbb{K}$ and compute the Hilbert series of$L_T^\mathbb{K}(2)$. Finally, we present concrete combinatorial formulas toobtain the Krull dimension of $S/L_T^\mathbb{K}(2)$ as well as lower and upperbounds for Krull dimension.

Gr\"obner basis and Krull dimension of Lov\'asz-Saks-Sherijver ideal associated to a tree