Capacitor between emitter and collector
For a circuit to oscillate it is necessary to have a LOOP GAIN of unity (Barkhausen criterion).
Hence, the amount of output voltage that is fed back to the input of the amplifying unit depends on the gain (magnitude and phase!).
In the present case, only a part of the output voltage is created across the impedance seen at the emitter node because of the voltage division rule.
This emitter node impedance is \$re=r6||(1/gm)\$ with \$gm=transconductance\$.

For the circuit shown, the exact calculation of loop gain is rather involved since the feedback circuit (C5-re) acts as an additional load for the transistor and, hence, must be considered for gain calculation (in parallel to the LC tank).
Something similar applies to the phase. 
Because the feedback path (high-pass C5-re) creates a certain phase shift (the capacitor C5 CANNOT be regarded as a short for "high frequencies") the resulting oscillation frequency will NOT be identical to the resonant frequency of the LC tank. Instead, there will be a small frequency shift.