Summary: In this article we study a quotient ring (the thinned algebra introduced by S. Sidki) of the group algebra over finite fields – in characteristic \(p\) – of an infinite class of Burnside \(p\)-groups acting on trees, of branch type. These groups are just-infinite and we prove, as has been shown by S. Sidki to hold for the Gupta-Sidki 3-group, the corresponding rings are also just-infinite. In most cases we are able to establish that they are also primitive. In the case of the Grigorchuk 2-group, we prove that its thinned algebra is either primitive or the augmentation ideal is nil, and furthermore, we show that it is abundant in nilpotent elements.