lec25.md

Advanced Trees, including Geometric

Traversals

Rooted Trees

• Tree iteration = tree traversal
• Unlike lists, many natural orderings

Tree Traversal

• Level Order
  • Traverse top-to-bottom, left-to-right
  • Nodes "visited" in given order
• Depth First Traversals
  • Preorder (root, left, right), Inorder (left, root, right), Postorder (left, right, root)
    • A Weird Trick
    • Preorder Traversal Runtime
• Level Order Traversal
  • Iterative Deepening
    • Runtime $$\Theta(N)$$ b/c each new level doubles amount of work done & # of nodes visited
      • Exponential in height of tree, $$\Theta(2^{H})$$ → height logarithmic in # of nodes, $$H = \Theta(\log{N})$$ → overall runtime linear in # of nodes, $$\Theta(N)$$
    • Spindly Runtime $$\Theta(N^{2})$$
    • Tree Height & Runtime

Range Finding

• Instead of traversing entire tree, only want to look at items in certain range

Pruning & findInRange Runtime

• Pruning: Restrict search to only nodes containing path(s) to answer(s)
• Runtime

Spatial Trees

Handling Multidimensional Data: Quadtrees

• Generalization of BST
• Think of items as oriented in particular 2D direction (e.g NW/NE/SW/SE instead of binary </>)
• Can have mutliple different quadtrees for same set of data
  • Just like BST, insertion order determines topology of QuadTree
• If on boundary line, usually assume = → >
• Pruning
  • Analogous to binary search on BST (eliminate unnecessary search spaces)
  • Ignore branches if no possible way branch could contain wanted item

Tree Iterators (Extra)

• Use Stack based DFS traversal