In Massechet Erchin 10b we are taught that we are obligated to say hallel on all yomim tovim, but on Succos, there is an additional unique requirement to say hallel; the recitation of hallel on Succos has a dual role in halacha.
There is a definite link between between the mitzvah of daled minim and kerias hallel. The incorporation of shaking the lulav within the context of hallel clearly demonstrates that there is a connection between these two mitzvos.
Chazal teach us that there is another mitzvah which is performed together with the recitation of hallel. The mishna in Masseches Pesachim teaches us that in the time of the Beis HaMikdash the korban pesach was offered simultaneously with the singing of hallel. Why are the mitzvos of daled minim and korban pesach singled out as mitzvos to be performed in conjunction with hallel?
The Gemara, in Masseches Megillah , relates that at the time of the miracle of Purim, Chazalwere unsure as to how to commemorate the event appropriately. They wanted to institute the reading of the megillah as an eternal commemoration of the miraculous events that had transpired, but were concerned with this being considered an "addition to the Torah." Ultimately, they delved into the Torah itself to find a source obligating the commemoration of a miracle. Chazal concluded that just as the Jewish people celebrated the culmination of Yetzias Mitzrayim by praising Hashem at Keriat Yam Suf with the singing of "Az Yashir", so too the events of Purim should be celebrated by praising Hashem, the Megillah acting as the vehicle of praise. It is this requirement of praising Hashem following deliverance from slavery that obligates us to recite hallel as we offer the korban pesach- the korban that transformed us from being slaves to being free men.
Chazal noted that there is a greater cause for celebration following the miracle of Purim: At Keriat Yam Suf it was freedom that was at stake, at Purim it was life itself that was threatened. They concluded that the occurrence of a miracle requires us to respond by praising Hashem. A miracle that ensures our freedom warrants celebration, how much more so for one that saves our very lives. The obligation to celebrate our being saved from death is the root of our hallel as we take the lulav on succos. On Rosh Hashana and Yom Kippur we did not say hallel. Chazal explain that as we are being judged and our lives are hanging in the balance it is inappropriate to sing the words of hallel. Following Yom Kippur, when we believe that we have been sealed in the sefer hachaim, we have in been delivered from the brink of death. The medrash compares the Jew who shakes a lulav on Succos to the warrior returning home victorious waving his spear to indicate that he has returned safely. As we lift our lulav in celebration of the gift of life we were granted on Yom Kippur; our entire being bursts forth in song, praising Hashem who has delivered us from death to life.