The maximum number of nodes at level $L$ of a binary tree is given by the formula $2^L$.

In a binary tree, each node can have at most two children, commonly referred to as the left child and the right child. The level of a node is determined by the number of edges or steps from the root node. Specifically, the root node is located at level $0$, its children are at level $1$, the children of those nodes are at level $2$, and this pattern continues accordingly.

The maximum number of nodes at any given level $L$ can be calculated using the formula $2^L$. This arises from the fact that each node can have up to two children. For instance, at level $0$ (the root), there can be $2^0 = 1$ node. At level $1$, there can be $2^1 = 2$ nodes. At level $2$, there can be $2^2 = 4$ nodes, and so forth.

This doubling pattern continues at each subsequent level. This behavior is characteristic of a specific type of binary tree known as a ‘complete binary tree’. In a complete binary tree, every level is fully filled, except possibly the last, and all nodes are positioned as far left as possible.

It is important to understand, however, that the number $2^L$ represents the maximum number of nodes at level $L$. A binary tree may contain fewer nodes at any level, depending on the number of children each node has. For example, if a node has only one child, then the number of nodes at the next level will be less than the maximum.

In summary, while the maximum number of nodes at level $L$ of a binary tree is $2^L$, the actual number of nodes can be lower depending on the tree’s structure. This concept is fundamental to understanding the properties and behaviors of binary trees, a crucial topic in computer science.