Whenever you are attempting to explain Bitcoin to anyone, it makes a lot of sense to simply state that the derivation private key > public key > address is deterministic. However, if you are interested in writing your own code or studying the existing code base from various libraries, you may wish to know that a private key can actually lead to two public keys (one in compressed form and another in uncompressed form), both being encodings of the same point on the secp256k1 elliptic curve but leading to two different hash values. Hence, we have two possible hash values and each hash value can in turn lead to two different (pay-to-public-key-hash) addresses (one for the main bitcoin network and one for the testing network). So all in all (from a developer perspective rather than a mere user of Bitcoin), a private key can lead to 4 possible addresses. I attach a java snippet:

import java.math.BigInteger;
import org.bitcoinj.core.ECKey;
import org.bitcoinj.core.NetworkParameters;
import org.bitcoinj.core.Address;
import org.bitcoinj.params.MainNetParams;
import org.bitcoinj.params.TestNet3Params;

public class Test {

  public static void main(String[] args){

  // An example of private key from the book 'Mastering Bitcoin'
  String k = "1E99423A4ED27608A15A2616A2B0E9E52CED330AC530EDCC32C8FFC6A526AEDD";

  // Converting our string encoding as an actual number
  BigInteger priv = new BigInteger(k,16);

  // Creating a key object from our private key, with compressed public key
  ECKey k1 = ECKey.fromPrivate(priv, true);

  // Creating a key object from our private key, with uncompressed public key
  ECKey k2 = ECKey.fromPrivate(priv, false);


  // 03f028892bad7ed57d2fb57bf33081d5cfcf6f9ed3d3d7f159c2e2fff579dc341a
  System.out.println(k1.getPublicKeyAsHex()); // compressed

  // 04f028892bad7ed57d2fb57bf33081d5cfcf6f9ed3d3d7f159c2e2fff579dc341a...
  //...07cf33da18bd734c600b96a72bbc4749d5141c90ec8ac328ae52ddfe2e505bdb
  System.out.println(k2.getPublicKeyAsHex()); // uncompressed

  NetworkParameters main = MainNetParams.get();   // main bitcoin network
  NetworkParameters test = TestNet3Params.get();  // test bitcoin network

  Address addr1 = k1.toAddress(main); // main network, compressed
  Address addr2 = k1.toAddress(test); // test network, compressed
  Address addr3 = k2.toAddress(main); // main network, uncompressed
  Address addr4 = k2.toAddress(test); // test network, uncompressed

  System.out.println(addr1.toString()); // 1J7mdg5rbQyUHENYdx39WVWK7fsLpEoXZy
  System.out.println(addr2.toString()); // mxdivjAqQSQj4LrAMX1XLQidyfU3pCWeS7
  System.out.println(addr3.toString()); // 1424C2F4bC9JidNjjTUZCbUxv6Sa1Mt62x
  System.out.println(addr4.toString()); // miY1V5L3QDaZVjrMT2Sw2WhHn63GzsNFQB

  }
}

You can generate unlimited number of addresses from the same private key. Private key is simply a number. from which you can generate compressed and uncompressed public keys. from the public keys you can generate addresses for each network (test, main, reg) for different types of altcoins by changing network byte.