You are to write a function min$\_$dist$\_$tree$($graph$,$ source$)$ that takes in a connected, weighted graph and a vertex as its input, and then returns the minimum distance tree produced by Dijkstra's Algorithm with 'source' as its root. Note that since the input graph will be connected, the resulting tree will be spanning.

print$($min$\_$dist$\_$tree$(\{"$A$"$ : $[["$B$",10],["$D$",5]],"$B$"$ : $[["$A$",10],["$C$",5]],"$C$"$ : $[["$B$",5],["$D$",15]],"$D$"$ : $[["$C$",15],["$A$",5]]\},"$A$"))$

This should return

$\{"$A$"$:$["$B$","$D$"],"$B$"$:$["$C$"],"$C$"$:$[],"$D$"$:$[]\}$