Sibling nodes in a binary tree are defined as nodes that share the same parent node.

In a binary tree, each node can have a maximum of two children, which are typically referred to as the left child and the right child. When two nodes share the same parent, they are classified as sibling nodes. For instance, consider a binary tree with a root node denoted as $A$, which has two child nodes $B$ and $C$. In this case, nodes $B$ and $C$ are siblings because they both have the same parent, $A$.

The concept of sibling nodes is crucial for understanding the structure and traversal of binary trees. Nodes in a binary tree are organized into levels, beginning with the root node at level $0$. While all nodes at the same level may share certain characteristics, they are not necessarily siblings, as they might not share the same parent. For example, in a binary tree with root $A$, children $B$ and $C$, and grandchildren $D$, $E$, $F$, and $G$ (where $D$ and $E$ are children of $B$, and $F$ and $G$ are children of $C$), the nodes $D$ and $E$ are siblings, as are $F$ and $G$. However, $D$ and $F$ are not siblings, even though they are at the same level.

The sibling relationship also plays a significant role in various operations performed on binary trees. For instance, in the context of tree rotations used in balancing operations of self-balancing binary search trees, sibling nodes are particularly important. During a right rotation, the left child of a node becomes the new parent of that node, while the original node takes on the role of the right child of its former left child. Furthermore, the original right child of the node becomes the left child of its former sibling.

In conclusion, a solid understanding of sibling nodes is essential for the study of binary trees. This knowledge aids in grasping the tree’s overall structure, the methods of traversal, and the various operations that can be performed on the tree.