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There's no point in limiting your function inputs to a vector. You can take two input iterators instead. It'll make your function more flexible and more "standard library-like".

One function should do one focused thing. Your function finds such an element and prints it as the same time. These concerns are unrelated. You can return std::optional<T> instead to represent either such an element or its absence. Note: it works for C++17 only. You can use boost::optional or return a smart pointer if it's not available.

There's no need for things like this anymore:

 typename std::map<T, int>::iterator itr;
 for(itr = count.begin(); itr != count.end(); itr++)

for (const auto& kv : count) is much better, isn't it?

You can also use initialization list to create a vector with the elements you need: std::vector<char> v({'r', 't',..., 's'})

for (T x : vec) creates an unnecessary copy. Use for (const T& x : vec) or for (const auto& T : vec) instead.

You can also use std::unordered_map to do the counting and then choose the smallest element (it's more efficient because one can find the minimum in O(n). It's "easier" than sorting the input). One caveat: you need a hash function for T in this case. You can pass it as another argument, defaulted to std::hash.

There's no point in limiting your function inputs to a vector. You can take two input iterators instead. It'll make your function more flexible and more "standard library-like".

One function should do one focused thing. Your function finds such an element and prints it as the same time. These concerns are unrelated. You can return std::optional<T> instead to represent either such an element or its absence. Note: it works for C++17 only. You can use boost::optional or return a smart pointer if it's not available.

There's no need for things like this anymore:

 typename std::map<T, int>::iterator itr;
 for(itr = count.begin(); itr != count.end(); itr++)

for (const auto& kv : count) is much better, isn't it?

You can also use initialization list to create a vector with the elements you need: std::vector<char> v({'r', 't',..., 's'})

for (T x : vec) creates an unnecessary copy. Use for (const T& x : vec) or for (const auto& T : vec) instead.

You can also use std::unordered_map to do the counting and then choose the smallest element (it's more efficient because one can find the minimum in O(n). It's "easier" than sorting the input). One caveat: you need a hash function for T in this case. You can pass it as another template parameter, defaulted to std::hash.