Graph burning is motivated by the spread of social influence, and the burning number measures the speed of the spread. Given that the smallest burning number among the spanning trees of a graph determines the burning number of a connected graph, trees are the main objects of investigation in graph burning. Given a prescribed burning number, our study focuses on identifying the corresponding extremal trees with respect to order up to graph homeomorphism. In this work, we propose the concept of admissible sequences over a homeomorphically irreducible tree in addition to developing a general framework. We then determine whether an admissible sequence induces an extremal tree with a specified burning number. Additionally, we obtain some results on the smallest attainable diameter for extremal $n$-spiders with a prescribed burning number.