This paper introduces Algebra B ( B for...Baskets !) , a structured framework for working with hierarchical objects called basket trees. A basket tree is a rooted tree whose nodes carry fixed capacities that strictly decrease from parent to child, while the internal content of each node can change freely. This separation between immutable capacity and mutable content creates a combinatorial setting that differs from classical arithmetic and from existing theories of labelled trees. The paper develops the basic operations on basket trees, including content insertion, content removal, and uniform enlargement of capacities. It then studies morphisms that describe how one basket tree can be embedded into another, and uses these embeddings to define a new form of division into quotient and remainder based on the number of disjoint occurrences of a given pattern. Several structural theorems are established, showing how depth, width, and capacity distributions behave under these operations. A central result is the construction of canonical configurations: for any valid collection of capacities, there exists a unique tree of minimal depth built by a simple greedy procedure. The framework also provides a characterization of which capacity profiles can be realized, and a classification of basket trees up to isomorphism. Taken together, these results show that Algebra B forms a coherent combinatorial theory with its own notions of structure, division, and canonical form, distinct from existing models in tree combinatorics, operads, and resource semantics.