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First Childnext Sibling RepresentationEdit

First Child/Next Sibling Representation, often shortened to FCNS, is a practical way to encode a rooted, ordered tree whose nodes can have any number of children using the binary-tree framework. In this scheme, each node stores two pointers: one to its first child and one to its next sibling. This compact encoding makes it easy to reuse binary-tree traversal code and data structures while still representing general trees found in many domains, from compilers to file systems and data interchange formats.

The FCNS representation is commonly described as a variant of the Left-Child Right-Sibling model, and it is discussed in many treatments of Tree data structure concepts and Binary tree implementations. It allows an arbitrary-arity tree to be navigated with binary-tree primitives, and it dovetails with a variety of algorithms used for Tree traversal and related operations. For more context on how this fits into broader hierarchy representations, see discussions of n-ary tree structures and how they contrast with fixed-arity models.

Structure and semantics

  • Node layout: In FCNS, a node typically contains at least two pointers:

    • firstChild: a link to the node’s first child (if any)
    • nextSibling: a link to the node’s immediate next sibling (or null if none) Some implementations also include a parent pointer, which can simplify certain operations, but it is not strictly required for basic FCNS traversal. See how these pointers map onto a binary-tree node in discussions of Left-Child Right-Sibling representations.

  • How it represents an n-ary tree: The root’s firstChild points to its first child; that child’s nextSibling points to the second child, and so on. Each child’s own firstChild pointer continues to nest a subtree, so a node can have any number of children without storing a variable-size array of child pointers. This makes FCNS a memory-efficient way to encode trees with nodes that have highly variable fan-out.

  • Relationship to a binary tree: FCNS turns an n-ary tree into a binary tree by treating the left pointer as firstChild and the right pointer as nextSibling. This perspective is why many textbooks present FCNS under the umbrella of binary-tree techniques, even though the structure originally encodes a general tree. See Binary tree and Left-Child Right-Sibling for more on this mapping.

  • Traversal implications: To enumerate all children of a node, you walk the nextSibling chain starting from firstChild. To traverse the whole subtree in a depth-first order, you recursively visit each node and, for that node, walk its child chain in order. Typical pseudocode follows patterns found in Tree traversal references.

Basic operations

  • Preorder traversal (visit node, then its children in order)

    • Pseudocode example:
    • traverse(node):
      • if node is null, return
      • visit(node)
      • child = node.firstChild
      • while child is not null:
      • traverse(child)
      • child = child.nextSibling

  • Insertion of a new child

    • As the first child (insert at the head):
    • newChild.nextSibling = parent.firstChild
    • parent.firstChild = newChild
    • As the last child (append at end): walk to the last node in the firstChild/nextSibling chain and link the new node there.
    • These operations illustrate the trade-off between simplicity (inserting at the head) and preserving a desired child-order (which may require a tail traversal).

  • Deletion of a child

    • Requires locating either the child’s previous sibling pointer or the previous pointer that references the child (often via parent traversal or a stored parent/prev reference).
    • Then adjust the relevant nextSibling or firstChild pointers to bypass the deleted node, and deallocate if needed.

  • Accessing the i-th child

    • In FCNS, there is no direct array-like access to the i-th child; you must walk i-1 nextSibling links starting from firstChild. This makes random access slower compared with certain array-based representations, but it preserves a compact, uniform structure.

Complexity and performance

  • Space: Each node stores two pointers (plus any payload). This is typically more space-efficient than maintaining a dynamic array of child pointers for nodes with many children, especially when many nodes have few children.
  • Time for operations:
    • Accessing the k-th child: O(k) in the worst case.
    • Enumerating all children of a node with k children: O(k).
    • Insertion at the head of a node’s child list: O(1).
    • Insertion at the tail: O(k) to reach the tail, unless a tail pointer is maintained.
    • Deletion of a known child: O(1) to adjust pointers if you have a direct reference to the prior node; otherwise, O(k) to locate it via traversal.
  • Traversal efficiency: Depth-first traversals map cleanly to standard binary-tree traversal algorithms, which makes FCNS attractive for compiler-backed structures such as Abstract syntax trees and other tree-structured representations.

Comparisons and trade-offs

  • With fixed-arity trees or when quick random access to a node’s specific child is essential, arrays or linked lists of children may offer faster direct access at the cost of more memory and more complex dynamic resizing.
  • FCNS excels when the tree is shallow but wide, or when memory locality is important and the cost of occasional linear scans is acceptable. It also harmonizes well with many classic binary-tree algorithms, making it a natural choice in educational contexts and in systems that reuse binary-tree tooling.
  • In modern data formats and DOM-like structures, FCNS or variants thereof are common because they support dynamic modification of the tree (insertion, deletion, reordering) without large restructuring overhead.

Applications and examples

  • Compilers and interpreters often build parse trees or intermediate representations that benefit from an FCNS encoding, enabling straightforward application of binary-tree traversal strategies to visit and transform subtrees. See Abstract syntax tree discussions for context.
  • XML and JSON processing can employ FCNS-style representations to model nested structures, with the firstChild link identifying the first child element and nextSibling connecting siblings at each level. See XML and Document Object Model discussions for related concepts.
  • File-system-like hierarchies, organizational charts, and other hierarchical data stores can be implemented using FCNS without requiring each node to carry an array of its children.

Example

A small rooted tree:

  • Root
    • Child1
    • Grandchild1
    • Grandchild2
    • Child2
    • Child3

In FCNS terms: - Root.firstChild -> Child1 - Child1.nextSibling -> Child2 - Child2.nextSibling -> Child3 - Child1.firstChild -> Grandchild1 - Grandchild1.nextSibling -> Grandchild2

Traversing from Root visits Root, then Child1, Grandchild1, Grandchild2, Child2, Child3 in a depth-first order, using the firstChild and nextSibling pointers.

See also