feat: Add Centroid Decomposition for trees (#7054)

- Implement CentroidDecomposition with O(N log N) construction
- Add CentroidTree class with parent tracking and query methods
- Include buildFromEdges helper for easy tree construction
- Add comprehensive test suite with 20+ test cases
- Cover edge cases, validation, and various tree structures

Closes #7054

Author: saikirankanakala

src/main/java/com/thealgorithms/tree/CentroidDecomposition.java

@@ -0,0 +1,212 @@
+package com.thealgorithms.tree;
+
+import java.util.*;
+
+/**
+ * Centroid Decomposition is a divide-and-conquer technique for trees.
+ * It recursively partitions a tree by finding centroids - nodes whose removal
+ * creates balanced subtrees (each with at most N/2 nodes).
+ * 
+ * Time Complexity: O(N log N) for construction
+ * Space Complexity: O(N)
+ * 
+ * Applications:
+ * - Distance queries on trees
+ * - Path counting problems
+ * - Nearest neighbor searches
+ * 
+ * @author lens161
+ */
+public final class CentroidDecomposition {
+
+    private CentroidDecomposition() {
+        // Utility class
+    }
+
+    /**
+     * Represents the centroid tree structure.
+     */
+    public static class CentroidTree {
+        private final int n;
+        private final List<List<Integer>> adj;
+        private final int[] parent;
+        private final int[] subtreeSize;
+        private final boolean[] removed;
+        private int root;
+
+        /**
+         * Constructs a centroid tree from an adjacency list.
+         * 
+         * @param adj adjacency list representation of the tree (0-indexed)
+         * @throws IllegalArgumentException if tree is empty or null
+         */
+        public CentroidTree(List<List<Integer>> adj) {
+            if (adj == null || adj.isEmpty()) {
+                throw new IllegalArgumentException("Tree cannot be empty or null");
+            }
+            
+            this.n = adj.size();
+            this.adj = adj;
+            this.parent = new int[n];
+            this.subtreeSize = new int[n];
+            this.removed = new boolean[n];
+            Arrays.fill(parent, -1);
+            
+            // Build centroid tree starting from node 0
+            this.root = decompose(0, -1);
+        }
+
+        /**
+         * Recursively builds the centroid tree.
+         * 
+         * @param u current node
+         * @param p parent in centroid tree
+         * @return centroid of current component
+         */
+        private int decompose(int u, int p) {
+            int size = getSubtreeSize(u, -1);
+            int centroid = findCentroid(u, -1, size);
+            
+            removed[centroid] = true;
+            parent[centroid] = p;
+            
+            // Recursively decompose each subtree
+            for (int v : adj.get(centroid)) {
+                if (!removed[v]) {
+                    decompose(v, centroid);
+                }
+            }
+            
+            return centroid;
+        }
+
+        /**
+         * Calculates subtree size from node u.
+         * 
+         * @param u current node
+         * @param p parent node (-1 for root)
+         * @return size of subtree rooted at u
+         */
+        private int getSubtreeSize(int u, int p) {
+            subtreeSize[u] = 1;
+            for (int v : adj.get(u)) {
+                if (v != p && !removed[v]) {
+                    subtreeSize[u] += getSubtreeSize(v, u);
+                }
+            }
+            return subtreeSize[u];
+        }
+
+        /**
+         * Finds the centroid of a subtree.
+         * A centroid is a node whose removal creates components with size ≤ totalSize/2.
+         * 
+         * @param u current node
+         * @param p parent node
+         * @param totalSize total size of current component
+         * @return centroid node
+         */
+        private int findCentroid(int u, int p, int totalSize) {
+            for (int v : adj.get(u)) {
+                if (v != p && !removed[v] && subtreeSize[v] > totalSize / 2) {
+                    return findCentroid(v, u, totalSize);
+                }
+            }
+            return u;
+        }
+
+        /**
+         * Gets the parent of a node in the centroid tree.
+         * 
+         * @param node the node
+         * @return parent node in centroid tree, or -1 if root
+         */
+        public int getParent(int node) {
+            if (node < 0 || node >= n) {
+                throw new IllegalArgumentException("Invalid node: " + node);
+            }
+            return parent[node];
+        }
+
+        /**
+         * Gets the root of the centroid tree.
+         * 
+         * @return root node
+         */
+        public int getRoot() {
+            return root;
+        }
+
+        /**
+         * Gets the number of nodes in the tree.
+         * 
+         * @return number of nodes
+         */
+        public int size() {
+            return n;
+        }
+
+        /**
+         * Returns the centroid tree structure as a string.
+         * Format: node -> parent (or ROOT for root node)
+         * 
+         * @return string representation
+         */
+        @Override
+        public String toString() {
+            StringBuilder sb = new StringBuilder("Centroid Tree:\n");
+            for (int i = 0; i < n; i++) {
+                sb.append("Node ").append(i).append(" -> ");
+                if (parent[i] == -1) {
+                    sb.append("ROOT");
+                } else {
+                    sb.append("Parent ").append(parent[i]);
+                }
+                sb.append("\n");
+            }
+            return sb.toString();
+        }
+    }
+
+    /**
+     * Creates a centroid tree from an edge list.
+     * 
+     * @param n number of nodes (0-indexed: 0 to n-1)
+     * @param edges list of edges where each edge is [u, v]
+     * @return CentroidTree object
+     * @throws IllegalArgumentException if n <= 0 or edges is invalid
+     */
+    public static CentroidTree buildFromEdges(int n, int[][] edges) {
+        if (n <= 0) {
+            throw new IllegalArgumentException("Number of nodes must be positive");
+        }
+        if (edges == null) {
+            throw new IllegalArgumentException("Edges cannot be null");
+        }
+        if (edges.length != n - 1) {
+            throw new IllegalArgumentException("Tree must have exactly n-1 edges");
+        }
+
+        List<List<Integer>> adj = new ArrayList<>();
+        for (int i = 0; i < n; i++) {
+            adj.add(new ArrayList<>());
+        }
+
+        for (int[] edge : edges) {
+            if (edge.length != 2) {
+                throw new IllegalArgumentException("Each edge must have exactly 2 nodes");
+            }
+            int u = edge[0];
+            int v = edge[1];
+            
+            if (u < 0 || u >= n || v < 0 || v >= n) {
+                throw new IllegalArgumentException("Invalid node in edge: [" + u + ", " + v + "]");
+            }
+            
+            adj.get(u).add(v);
+            adj.get(v).add(u);
+        }
+
+        return new CentroidTree(adj);
+    }
+}