Yes, a rooted tree can technically be re-rooted at any node, creating a new valid rooted tree structure. The key constraint is that the underlying structure must still be a connected, acyclic graph with clearly defined parent-child relationships stemming from the chosen root. When a new root is chosen, all edges must be reoriented to reflect new parent-child relationships, and the hierarchical structure must be updated accordingly. For example, if a tree has node A as the current root and you re-root the tree at node C, C becomes the new origin, and the direction of edges must be inverted to maintain the rooted structure. However, in most practical applications, the root is chosen based on the context of the data—such as the top folder in a directory tree or the main category in a taxonomy. So, while multiple roots are possible by reinterpretation, typically only one root is used at a time to define the hierarchy.

Having exactly one parent per node (except the root) ensures that the structure of a rooted tree is unambiguous, predictable, and hierarchical. If a node had multiple parents, it would introduce complications such as cycles or multiple paths to a single node, violating the definition of a tree. One-parent-per-node guarantees that every node has a unique path back to the root, which is essential for many algorithms like traversal, searching, and recursion. This property is especially important in applications like file systems or expression evaluation, where understanding the precise origin and context of each node matters. For instance, in a family tree, if a child had multiple direct ancestors as "parents" from the same level, it would complicate generational analysis. In data processing, this uniqueness simplifies how resources are organised and accessed, as the system always knows where each element comes from and how to reach it efficiently.

The number of leaf nodes provides insight into how the tree branches and its overall shape. A tree with many leaf nodes relative to its total number of nodes tends to be wide and shallow, meaning it has more breadth than depth. Conversely, a tree with few leaf nodes is typically deeper, with longer chains of internal nodes leading to terminal nodes. This can help evaluate how balanced or skewed the tree is. Additionally, in rooted trees used in computing, such as in decision trees or syntax trees, the number of leaf nodes may indicate the number of possible outcomes or end states. For example, in a classification model, each leaf might represent a final decision. Understanding how many leaves exist helps in optimising traversal and search operations, and is also useful for assessing time complexity in certain algorithms. The relationship between internal and leaf nodes can also be formalised in some tree types to analyse structural efficiency.

When a node is deleted from a rooted tree, its impact on the tree depends on whether it is a leaf, internal node, or the root. If a leaf node is deleted, the operation is simple—its parent now has one fewer child, and the overall structure remains intact. If an internal node is deleted, the situation becomes more complex. In most implementations, one of the following actions is taken: (1) its children are reassigned to its parent, (2) the entire subtree rooted at the deleted node is removed, or (3) a suitable replacement node (such as one of its children) is promoted to its place. If the root node is deleted, a new root must be chosen, and all relationships must be restructured to reflect the new hierarchy. This may involve a full traversal and redefinition of parent-child links. Deleting nodes requires careful management to maintain the tree's acyclic and hierarchical properties.

A subtree in a rooted tree is defined as any node along with all of its descendants. Essentially, a subtree is itself a rooted tree, with its root being the node from which the subtree originates. Subtrees are useful because they allow complex trees to be broken down into manageable, self-contained components. This is especially important in recursive algorithms, where the problem is solved by working on a node and recursively solving problems on its subtrees. For instance, in a file directory, a folder and all its contents can be treated as a subtree. This abstraction is helpful in tasks like copying, searching, or deleting entire sections of data. In programming, trees are often processed by applying functions to each subtree. Understanding and leveraging subtrees makes reasoning about tree operations simpler and more modular. It also enables dynamic programming solutions and efficient computations, such as calculating the size or height of parts of the tree independently.