Uncountably many homogeneous real trees with the same valence

For any cardinal $\kappa \ge 2$, there is a unique complete real tree whose points all have valence $\kappa$. In this note, we show that, when $\kappa \ge 3$, it is necessary to assume completeness. More precisely, we show that there exist uncountably many homogeneous incomplete real trees whose points all have valence $\kappa$.