The concept of Gutman index \( Gut\left( G \right) \) of a connected graph \( G \) was introduced in 1994. The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. The Steiner Gutman k-index \( SGut_{k}\left( G \right) \) introduced by Mao et al. in 2018, is defined by \( SGut_{k}\left( G \right)=\sum_{S\subseteq V\left( G \right), |S|=k}\left[ \prod_{v \in S}d_{G}\left( v \right) \right]d_{G}\left( S \right), \) where \( d_{G}\left( S \right) \) is the Steiner distance of \( S \) and \( d_{G}\left( v \right) \) is the degree of \( v \) in \( G \). In this paper, we obtained the relations between Steiner Gutman \( k \)-index and Gutman index, Wiener index and Steiner Wiener \( k \)-index of trees with \( k=3, 4 \).