Custom Vector Template - Strings

I am attempting to read in a list of strings from the keyboard to a custom vector template that I wrote, however for some unknown reason, the program ALWAYS crashes after the second or third input when I attempt to write to or read the vector object.

Note: The entire point of the assignment is to write your own custom vector class and I am not allowed to use any feature provided by the C++ STL, such as arrays.

edit: Everything works without fault if I make it an int or char vector, but it needs to do all 3.

main.cpp

#include <iostream>
 #include "myvector.h"
 #include <iostream>
 #include <fstream>
 #include <sstream>
 #include <string.h>
 
 using namespace std;
 
 int main()
 {
     myvector<string> vec;
 
     string tmp;
 
     while(1) {
         cin >> tmp;
         if(tmp=="q"){
             break;
         }
         vec.push_back(tmp);
     }
 
     cout << "You have entered " << vec.size() << " elements. These are:" << endl;
 
     for(int i=0;i<vec.size();i++) {
         cout << vec[i] << endl;
     }
 
     return 0;
 }
 

myvector.h

#ifndef MYVECTOR_H
 #define MYVECTOR_H
 
 #include <iostream>
 #include <stdlib.h>
 #include <string.h>
 
 using namespace std;
 
 template<class T>
 class myvector
 {
     public:
         myvector();
         myvector(const myvector&);
         ~myvector();
         void push_back(const T&);
         int size();
         T operator[](int);
         myvector& operator=(const myvector&);
     protected:
     private:
         int vsize,maxsize;
         T* array;
         void alloc_new();
 };
 
 template<class T>
 myvector<T>::myvector()
 {
     maxsize=8;
     array=(T *)malloc(maxsize*sizeof(T));
 
     if (array == NULL) {
     cout << "Memory failure";
     }
     vsize=0;
 }
 
 template<class T>
 myvector<T>::myvector(const myvector& v)
 {
     maxsize=v.maxsize;
     vsize=v.vsize;
     array = (T *)malloc(maxsize*sizeof(T));
     for(int i=0;i<v.vsize;i++) {
         array[i]=v.array[i];
     }
 }
 
 template<class T>
 myvector<T>::~myvector()
 {
     free(array);
 }
 
 template<class T>
 void myvector<T>::push_back(const T& i)
 {
     if(vsize>=maxsize) {
         alloc_new();
     }
     array[vsize]=i;       //CRASHES KEEP HAPPENING HERE I THINK
     vsize++;
 }
 
 template<class T>
 T myvector<T>::operator[](int i)
 {
     return array[i];
 }
 
 template<class T>
 void myvector<T>::alloc_new()
 {
     maxsize=vsize*2;
     T* tmp=(T *)malloc(maxsize*sizeof(T));
     for(int i=0;i<vsize;i++) {
         tmp[i]=array[i];
     }
     free(array);
     array=tmp;
 }
 
 template<class T>
 int myvector<T>::size()
 {
     return vsize;
 }
 
 template<class T>
 myvector<T>& myvector<T>::operator=(const myvector& v)
 {
     if(this!=&v) {
         maxsize=v.maxsize;
         vsize=v.vsize;
         delete[] array;
         array = (T *)malloc(maxsize*sizeof(T));
         for(int i=0;i<v.vsize;i++) {
             array[i]=v.array[i];
         }
     }
     return *this;
 }