quantum mechanics - Number of Nodes in energy eigenstates

I have a question from the very basics of Quantum Mechanics. Given this theorem:

For the discrete bound-state spectrum of a one-dimensional potential let the allowed energies be $E_1

What is the physical interpretation for the number of nodes in the concrete energy eigenstate? I understand that the probability of finding the particle in the node point is $0$ for the given energy. However, why does the ground state never have a node? or why does every higher energy level increments number of nodes precisely by 1?

Answer

I guess there is not that much to grasp, unless you can really understand dark spots on an electron diffraction pattern. Very roughly explanation would be to interpret wave functions of a particle in a potential well as "standing waves", or as two interfering waves reflected from the walls of the well. Increasing the energy leads to higher harmonics, which leads to additional nodes. Nodes' numbering is the same as in the case of a classical string.