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A classification tree, also known as a decision tree for classification, is a machine learning algorithm used for solving classification problems. It is a type of supervised learning technique that is primarily employed for predicting categorical labels or class memberships for input data points.
Here's how a classification tree works:

Tree Structure: Similar to a regression tree, a classification tree consists of a treelike structure where each node represents a decision or a splitting point based on one of the input features. The tree starts with a root node and branches into multiple child nodes based on specific criteria.

Splitting Criteria: At each internal node (nonleaf node) of the tree, a decision is made about which feature and which threshold value should be used to split the data into two or more subsets. The goal is to find the feature and threshold that maximize the separation or purity of classes within each subset. Common measures of separation or purity include Gini impurity, entropy, or classification error rate.

Leaf Nodes: The splitting process continues recursively until a stopping criterion is met. This criterion could be a predefined depth limit, a minimum number of data points in a node, or a purity threshold. When the stopping criterion is met, the node becomes a leaf node, and it contains the predicted class label for the input data. This prediction is typically the majority class label within that leaf node's subset of data.

Prediction: To make predictions for new data points, you traverse the tree from the root node down to a leaf node, following the decision rules at each node. When you reach a leaf node, you use the predicted class label stored in that leaf node as the final prediction for the input data.
Classification trees are used in a wide range of applications, including spam email classification, medical diagnosis, customer churn prediction, and more. They are particularly attractive because they are easy to interpret and can capture complex decision boundaries in the data. However, like regression trees, classification trees are also prone to overfitting, especially when the tree becomes too deep and complex. To address overfitting, techniques like pruning and ensemble methods such as Random Forests or Gradient Boosting are often employed.
The key difference between classification trees and regression trees lies in the nature of the response variable they are designed to predict:

Classification Trees:
 Response Variable: Classification trees are used for predicting categorical or discrete class labels. The response variable in classification trees represents categories or classes. Examples of classification tasks include spam detection (classifying emails as spam or not spam), disease diagnosis (classifying patients into disease categories), and sentiment analysis (classifying text as positive, negative, or neutral).
 Node Outputs: In a classification tree, the leaf nodes (end nodes) contain the predicted class label for the input data point. The prediction is typically the majority class label among the data points that reach that leaf node during the tree traversal.
 Regression Trees:
 Response Variable: Regression trees are used for predicting a continuous numeric output variable. The response variable in regression trees represents a realvalued quantity. Examples of regression tasks include predicting house prices, estimating a person's age based on certain features, or forecasting stock prices.
 Node Outputs: In a regression tree, the leaf nodes contain the predicted numeric value for the input data point. The prediction is typically the mean, median, or some other statistical measure of the target variable values among the data points that reach that leaf node.
Note that decision trees can be used for both classification and regression tasks, and they are capable of handling both categorical and numerical variables.
Furthermore, decision trees are fairly high variance models. (see page4313)
Table 4002. Applications and related concepts of decision tree.
