feat: add -tree chapter

Author: aaronoah

docs/SUMMARY.md

@@ -14,6 +14,7 @@
     * [Overview][Randomization Overview]
     * [Discrete Probability](randomization/discrete-probability.md)
     * [Randomized Algorithms](randomization/randomized-algorithms.md)
+    * [Reservoir Sampling](randomization/reservoir-sampling.md)
 * [Sorting][Sorting Overview]
     * [Overview][Sorting Overview]
     * [Selection Sort](sorting/selection-sort.md)
@@ -23,6 +24,7 @@
     * [Merge Sort](sorting/merge-sort.md)
     * [Heap Sort](sorting/heap-sort.md)
     * [Quick Sort](sorting/quick-sort.md)
+    * [Non-Comparison Sort](sorting/non-comparison-sort.md)
 * [Graph Algorithms][Graph Algo]
     * [Overview][Graph Algo]
     * [Minimum Spanning Tree](graph-algorithms/minimum-spanning-tree.md)
@@ -37,6 +39,7 @@
     * [Heap](tree/heap.md)
     * [Disjoint Set](tree/disjoint-set.md)
     * [Threaded Binary Tree](tree/threaded-binary-tree.md)
+    * [Segment Tree](tree/segment-tree.md)
 * [Searching][Searching Overview]
     * [Overview][Searching Overview]
     * [Binary Search](searching/binary-search.md)

docs/images/segment-tree.png

@@ -12,5 +12,6 @@ In the algorithm design, [randomization](#randomization) plays a pivot role in p
 
 * [Discrete Probability](discrete-probability.md) - probability analysis, indicator random variable and several statistics definitions covered.
 * [Randomized Algorithms][randomized] - median-finding problem, i<sup>th</sup> order statistic finding problem are analyzed and solved randomly and deterministic.
+* [Reservoir Sampling](reservoir-sampling.md) - an introduction to a common technique in data processing.
 
 [randomized]: randomized-algorithms.md

docs/randomization/overview.md

@@ -42,3 +42,4 @@ A _**complete k-ary tree**_ is a tree in which all leaves have the same _depth_
 * [B-Tree](b-tree.md) - a disk based tree structure, particularly useful in database design.
 * [Disjoint-Set](disjoint-set.md) - a set operation that finds relevance among data and build up forests for faster manipulations.
 * [Threaded Binary Tree](threaded-binary-tree.md) - a special form of binary tree that connect leaves to parents, useful in designing Morris traversal, which require Ο(1) extra space.
+* [Segment Tree](segment-tree.md) - a binary tree used for storing intervals or segments within a data array to speed up queries such as [Range Minimum Query](segment-tree.md).

docs/tree/overview.md

@@ -0,0 +1,21 @@
+# Segment Tree
+
+Segment tree is a contextual binary tree designed for storing intervals or segments of a series of data items to speed up specific queries such as [Range Minimum Query](https://en.wikipedia.org/wiki/Range_minimum_query). Being contextual, each node of the segment tree represents an interval of data indices. Consider the following example:
+
+An array of integers _INPUT_ of size N, build a segment tree _T_ which has:
+
+1. The _root_ of _T_ represents the data to be queried within interval _INPUT[0...N-1]_
+2. Leaves represent a single element INPUT[i] where 0 <= i < N.
+3. The internal nodes in level _h_  of _T_ represents 2<sup>h</sup> number of nodes, each have data to be queried from interval from _INPUT_[i...j], where 0 <= i < j < N
+
+There are N leaves in T and N - 1 internal nodes. Thus, the total number of nodes is 2 &sdot; N - 1. When the segment tree is built, its structure cannot be changed.
+
+<figure style="text:center">
+  <img src="../images/segment-tree.png" />
+  <figcaption>Figure 1. A Segment Tree Built on an Array of size 7</figcaption>
+</figure>
+
+There are two operations that can be performed on a segment tree:
+
+1. update: given an index and a value within the array, update the corresponding element. The time complexity is &Omicron;(log N) time.
+2. query: given index _l_ and _r_, return the contextual value within the interval from _l_ to _r_. (e.g. the minimal value within the range). In average, each query takes &Omicron;(log N) time.