555 saw tooth generator diodes purpose?
The PNP transistor is part of a constant current source for the capacitor. The current is determined by the voltage across \$R_E\$ (ignoring the relatively small base current). 

The voltage across \$ R_E\$ is \$ V_Z - V_{BE} + V_D \$ where \$V_D\$ is the forward voltage across the diode, so we can say that \$I_C \approx\$ \$ (V_Z - V_{BE} + V_D)\over R_E \$

Although the currents through the diode and the base-emitter junction are typically quite different, the voltages are not that far apart, and the magnitude of the temperature coefficients of both will be about 2mV/°C, and they will tend to cancel. 

In that case, \$I_C \approx\$  \$ V_Z\over R_E \$

The Zener voltage in this circuit must be less than 5V for this circuit to work (as the capacitor has to charge to 2/3 of 15V). The temperature coefficient of zeners in the 4.3-4.7V range is approximately zero.

(Ref Motorola Semiconductor Data Library 1N47xx series)

 

The combination of the zener diode and the diode stabilizes the constant current against both temperature changes and changes in the supply voltage, however it actually increases the variation in time with supply voltage since the threshold voltage of the 555 is derived by a divider from the supply voltage. 

Edit: In reference to an additional question, I include a "Functional Table" from a version of the 555 datasheet (CMOS type, but the logic is the same).